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The first part of this paper is an expanded version of a plenary lecture of the same title, given by the author at the CMFT conference at Bilkent University, Ankara, in June 2009. In the second part of the paper, a considerably stronger version of one of the main results is proved.
It is known that if f is a meromorphic map of the complement of three points on the extended complex plane into itself, then f is a Möbius map. We consider which subdomains of the extended complex plane also have this property.
We survey some results on the iteration of quasiregular mappings. In particular we discuss some recent results on the dynamics of quasiregular maps which are not uniformly quasiregular.
We investigate some relations between the reproducing kernels of Hilbert spaces of holomorphic and harmonic functions on the unit balls and the radial differential operators acting on the spaces that allow their characterization via integrals of their derivatives on the balls. We compare and...
This article provides an overview of the properties and uses of the Schottky-Klein prime function on the Schottky double of multiply connected planar domains. Simple expressions are offered for the conformal mappings from a multiply connected circular domain to canonical multiply connected slit...
In this paper we survey recent results about Fischer decompositions of polynomials or entire functions and their applications to holomorphic partial differential equations. We discuss Cauchy and Goursat problems for the polyharmonic operator. Special emphasis is given to the Khavinson-Shapiro...
We present a brief introduction to logarithmic capacity, and describe in detail a method for computing it using quadratic minimization. The method yields rigorous upper and lower bounds, which can in principle be made arbitrarily close to one another.
A special infinite series of real numbers which appeared in a comic strip is summed. Some of the techniques used are standard in complex calculus, but are now applied in a different context. Two other basic problems where complex numbers are useful are described. One, the solution of cubics is...
We will describe a method for proving that a given real number is irrational. It amounts to constructing explicit rational approximants to the real number which are “better than possible” should the real number be rational. The rational approximants are obtained by evaluating a Hermite-Padé...
It is possible to approximate the Riemann zeta-function by meromorphic functions which satisfy the same functional equation and satisfy (respectively do not satisfy) the analogue of the Riemann hypothesis.
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